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is algebraic; it is the root of 99900 x - 1 = 0 (as may be verified
by the reader who remembers how to sum an infinite convergent
geometrical progression).

The first to prove that certain numbers are transcendental
was Joseph Liouville (the same man who encouraged Hermite
to write to Jacobi) who, in 1844, discovered a very extensive
class of transcendental numbers, of which all those of the form

where n is a real number greater than 1 (the example given
above corresponds to n *= 10), are among the simplest. But it
is probably a much more difficult problem to prove that a
particular suspect, like e or K, is or is not transcendental than
it is to invent a whole infinite class of transcendentals: the
inventive mathematician dictates - to a certain extent - the
working conditions, while the suspected number is entire
master of the situation, and it is the mathematician in this case,
not the suspect, who takes orders which he only dimly under-
stands. So when Hermite proved in 1873 that e (denned a short
way back) is transcendental, the mathematical world was not
only delighted but astonished at the marvellous ingenuity of
the proof.
Since Hermite's time many numbers (and classes of numbers)
have been proved transcendental. What is likely to remain a
high-water mark on the shores of this dark sea for some time
may be noted in passing. In 1934 the young Russian mathe-
matician Alexis Gelf ond proved that all numbers of the type a6,
where a is neither 0 nor 1 and b is any irrational algebraic
number, are transcendental. This disposes of the seventh of
David Hilbert's list of twenty-three outstanding mathematical
problems which he called to the attention of mathematicians
at the Paris International Congress in 1900. Note that ^irra-
tionaP is necessary in the statement of Gelfond's theorem (if
b  w/7?i, where n, m are rational integers, then a*, where a is
any algebraic number, is a root of xm  an = 0, and it can be
shown that this equation is equivalent to one in which all the
coefficients are rational integers.